Critical Path Method for Scheduling

Implement the Critical Path Method (CPM) for project scheduling. Given a set of tasks with durations and dependency relationships forming a directed acyclic graph (DAG), compute: 1. Project duration : The minimum total time to complete all tasks. 2. Critical path : The sequence of tasks (in execution order) that determines the project duration. These are tasks with zero slack, meaning any delay in them delays the entire project. 3. Slack : For each task, the amount of time it can be delayed without affecting the project duration. Your function receives: tasks: A dictionary mapping task names (strings) to their durations (integers). dependencies: A dictionary mapping each task name to a list of prerequisite task names that must be completed before it can start. Tasks with no prerequisites have an empty list. The function should return a dictionary with three keys: 'project duration': An integer representing the minimum time to finish all tasks. 'critical path': A list of task names on the critical path, ordered by execution sequence. When multiple tasks have the same topological precedence, use alphabetical ordering. 'slack': A dictionary mapping each task name to its total slack (integer). Assume the input always forms a valid DAG with no cycles.